# Glueing sheaves on Grothendieck sites

Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf $F:\mathsf{Open}(X)^{\mathrm{op}}\to \mathsf{Sets}$, iff the family satisfies some local compatibility conditions like the cocycle condition on threefold unions. This is the situation for sheaves on a topological space.

Is there a similar statement for sheaves on a Grothendieck site $C$, in particular for the site $\mathsf{Sch}$ of schemes with the Zariski- or the etale topology?

Perhaps there is a sheaf $F:C^{\mathrm{op}}\to \mathsf{Sets}$ iff there is a compatible family $F:C^{\mathrm{op}}/V_i\to \mathsf{Sets}$ of sheaves on the slice topoi? What are the compatibility conditions then? Does somebody have a reference?

My second question is most probably nonsense but it would be nice if somebody could support this such that I can forget about this idea forever, even for the situation of sheaves on a topological space: Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ and $G_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be two families of sheaves and $F_i\to G_i$ a morphism of sheaves. Does this glue to a morphism $F\to G$ of the glued sheaves?

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I hope you don't mind, I have edited your question a bit so that the categories are given names that are formatted differently than normal math-mode symbols (although I'm sure no one would confuse $Open$ with the product of four symbols named "$O$", "$p$", "$e$", and "$n$", a visual distinction is always nice). Please feel free to undo my edit, or change it some more, if you don't like the way I changed it. –  Zev Chonoles Apr 8 '13 at 14:53
Dear @ZevChonoles, thank you for the editing. –  Ronald Bernard Apr 8 '13 at 17:11

Ok, so even the second question with the morphisms has a positive answer. I am surprised for the following reason: If you want to construct a morphism between two schemes, you can't do it locally. You can only check locally if a given morphism is an isomorphism. Even in topology, There are for example two non-isomorphic $2$-sheeted covering spaces! What is different in this situation? Sorry but I am confused now. –  Ronald Bernard Apr 8 '13 at 18:14